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Unformatted text preview: How to judge a series is convergent or divergent 1. Definition Given a series n=1 an = a1 + a2 + a3 + , let sn denote its nth partial sum:
n sn =
i=1 ai = a1 + a2 + + an If the sequence sn is convergent and limn sn = s exists as a real number, then the series an is called convergent and we write a1 + a2 + + an + = s or
n=1 an = s The number s is called the sum of the series. Otherwise, the series is called divergent. 2. The Test for Divergence If limn an does not exist or if limn an = 0, then the series divergent. n=1 an is 3. The Integral Test Suppose f is a continuous, positive, decreasing function on [1, ) and let an = f (n). Then the series an is convergent if and only if the improper n=1 integral 1 f (x)dx is convergent. 4. The Comparision Tests 4.1 The Comarision Test Suppose that an and bn are series with positive (i) If bn is convergent and an bn for all n, then (ii) If bn is divergent and an bn for all n, then 4.2 The Limit Comparision Test Suppose that an and bn are series with positive lim n an =c bn terms. an is also convergent. an is also divergent. terms. If where c is a finite number and c > 0, then either both series converge or both diverge. 1 5. The Alternating Series Test If the alternating series (1)n1 bn = b1  b2 + b3  b4 + bn > 0 n=1 satisfies (i) bn+1 bn for all n (ii) then the series is convergent. lim bn = 0 n 2 Important Series 1. Geometric Series n=1 arn1 = a + ar + ar2 + is convergent if r < 1 and its sum is n=1 arn1 = a 1r r < 1 If r 1, the geometric series is divergent. 2. Harmonic Series 1 1 1 = 1 + + + 2 3 n=1 n is divergent. 3. pseries The pseries 1 n=1 np is convergent if p > 1 and divergent if p 1. 3 11.2.30 ln(
n=1 n ) 2n + 5 4 11.3.20 n=1 4n2 1 +1 5 11.4.18 arctann n4 n=1 6 11.4.24 n2  5n 3 n=1 n + n + 1 7 11.5.10 (1)n1 n=1 2n2 4n2 + 1 8 ...
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 Spring '08
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 Calculus, Geometric Series

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